The geometry of permutation modules

TL;DR

This paper studies the structure of permutation modules over finite groups in positive characteristic, using stratification and cohomology to classify ideals within their derived categories.

Contribution

It introduces a stratification approach for permutation modules' derived categories and classifies their thick and localizing ideals using cohomological methods.

Findings

Spectrum of compact objects characterized via elementary abelian groups

Classification of thick and localizing ideals achieved

Use of twisted cohomology to describe the spectrum

Abstract

We consider the derived category of permutation modules for a finite group, in positive characteristic. We stratify this tensor triangulated category using Brauer quotients. We describe the spectrum of its compact objects, by reducing the problem to elementary abelian groups and then by using a twisted form of cohomology to express the spectrum locally in terms of the graded endomorphism ring of the unit. Together, these results yield a classification of thick and of localizing ideals.